Although the LC II and Classic II are often compared to the Color Classic, they are actually less related to it than the LC III is. Sure, the processor and memory may be handled the same, but the actual architecture seems to be more in line with the LC III (LC 520), LC III+ (LC 550), and LC 475 (LC 575). Proof to support this theory is the similarities between the LC 5xx and "Class of 1993" pizza box LCs.
More proof comes in the AppleDesign book, where there were originally three concepts of new all-in-ones, with one being the Color Classic, one being the LC 520, and the third being an intermediate that perhaps would have been around a IIsi in power.
Apple also was into selling one Mac under different labels and also had a somewhat universal board system (i.e. swapping a Color Classic board for an LC 550). My belief is that the Color Classic board is an updated version of the LC II, just as the LC II was a slightly modified Classic II (although it was newer, it COULD run straight-up 6.0.8, something the Classic II can't run...but that was probably for the stripped-down education models; the regular Classic was the compact in the education world around here and not the more business-oriented Classic II--which they tried to thrust into the SE/30's roster spot).
Since I believe all of these boards were designed around the same time and finished after the completion of System 7.1, there is, in my opinion, no conceivable way to run System 6 on them. They have their unique display connections, front bezel buttons, internal microphones, etc. that are unique to those boards and require the enablers to run. Since the routines needed to make these boards work correctly aren't in any 6.0.x, even 6.0.8L (which supports the Classic II), you won't be able to run anything without that code.